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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Sequences and Series

NDA 2016 · Mathematics · Sequences and Series

How many geometric progressions is/are possible containing 27,8 and...

How many geometric progressions is/are possible containing 27,8 and 12 as three of its/their terms?

Options

  1. AOne
  2. BTwo
  3. CFour
  4. DInfinitely many

Correct answer

D. Infinitely many

Step-by-step solution

Let ' aa ' be the first term \& ' xx ' be the common ratio. Also, suppose 27,8  and 1227,8 \ \text{ and } 12 be the pth ,qth   and rth \mathrm{p}^{\text{th }}, \mathrm{q}^{\text{th }} \ \text{ and } \mathrm{r}^{\text{th }} term of the G.P. ∴axp−1=27axq−1=8axr−1=12\begin{aligned} \therefore \quad \mathrm{ax}^{\mathrm{p}-1} & =27 \\ \mathrm{ax}^{\mathrm{q}-1} & =8 \\ & \mathrm{ax}^{\mathrm{r}-1} & =12 \end{aligned} Now, 27×82=12327 \times 8^2=12^3⇒axp−1x(axq−1)2=(axr−1)3⇒xp−1⋅x2q−2=x2r−3⇒p−1+2q−2=3r−3\begin{aligned} & \Rightarrow \quad \mathrm{ax}^{\mathrm{p}-1} \mathrm{x}\left(\mathrm{ax}^{\mathrm{q}-1}\right)^2=\left(\mathrm{ax}^{\mathrm{r}-1}\right)^3 \\ & \Rightarrow \quad \mathrm{x}^{\mathrm{p}-1} \cdot \mathrm{x}^{2 \mathrm{q}-2}=\mathrm{x}^{2 \mathrm{r}-3} \\ & \Rightarrow \mathrm{p}-1+2 \mathrm{q}-2=3 \mathrm{r}-3\end{aligned}⇒p+2q−3r=0\Rightarrow p+2 q-3 r=0 ...(1) There are infinitely many solutions for the eq. (1).
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